Solve the inequality of w
1/2w-1<-4w+7
Simplify as much as possible
To solve the inequality 1/2w - 1 < -4w + 7, you can follow these steps:
1. Simplify each side of the inequality:
- Add 4w to both sides: 1/2w + 4w - 1 < 7
- Combine like terms: 9/2w - 1 < 7
2. Add 1 to both sides: 9/2w < 8
- Now the inequality is: 9/2w < 8
3. Multiply both sides by 2/9 to isolate w:
(2/9) * (9/2w) < (2/9) * 8
- The inequality becomes: w < 16/9
So, the solution to the inequality 1/2w - 1 < -4w + 7 is w < 16/9.
A model rocket is launched with an initial upward velocity of 195ft/s. The rocket’s height h in feet after t seconds is given by the following
h=195t-16t^2
Find all the values of t for which the rocket height is 87feet
To find the values of t for which the rocket's height is 87 feet, you can set the equation h = 87 and solve for t.
87 = 195t - 16t^2
Rewriting the equation in standard form:
16t^2 - 195t + 87 = 0
To solve this quadratic equation, you can either factor it or use the quadratic formula. In this case, factoring may not be straightforward, so let's use the quadratic formula.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 16, b = -195, and c = 87. Plugging these values into the quadratic formula, we get:
t = (195 ± √((-195)^2 - 4 * 16 * 87)) / (2 * 16)
Simplifying under the square root:
t = (195 ± √(38025 - 55776)) / 32
t = (195 ± √(-17751)) / 32
Since √(-17751) is an imaginary number, there are no real solutions for t. Therefore, there are no values of t for which the rocket's height is exactly 87 feet.
Multiply
(5u-x+7)(3u-x)
To multiply (5u - x + 7)(3u - x), you can use the distributive property and combine like terms. Here's how:
First, distribute the 3u - x to each term in the first parenthesis:
(5u - x + 7)(3u - x) = (5u)(3u) + (5u)(-x) + (5u)(7) + (-x)(3u) + (-x)(-x) + (-x)(7)
Simplifying each term:
= 15u^2 - 5ux + 35u - 3ux + x^2 - 7x
Now, combine like terms:
= 15u^2 - 8ux + 35u + x^2 - 7x
So, the product of (5u - x + 7)(3u - x) is 15u^2 - 8ux + 35u + x^2 - 7x.
To solve the inequality 1/2w - 1 < -4w + 7, you need to isolate the variable w.
Let's start by simplifying both sides of the inequality:
1/2w - 1 < -4w + 7
First, let's eliminate the fraction by multiplying every term by 2:
2 * (1/2w - 1) < 2 * (-4w + 7)
This gives us:
w - 2 < -8w + 14
Now, let's collect like terms on each side:
w + 8w < 14 + 2
Combining like terms, we get:
9w < 16
Next, we want to isolate w by getting rid of the coefficient 9. To do that, we divide both sides of the inequality by 9:
(9w)/9 < (16)/9
Simplifying further, we have:
w < 16/9
So the solution to the inequality 1/2w - 1 < -4w + 7 is w < 16/9.
To solve the inequality 1/2w - 1 < -4w + 7, we need to isolate the variable w on one side of the inequality sign.
Let's start by simplifying the expression on both sides of the inequality:
1/2w - 1 < -4w + 7
To eliminate the fractions, we can multiply the entire inequality by the LCD (least common denominator), which in this case is 2. By doing so, we get rid of the fraction:
2 * (1/2w - 1) < 2 * (-4w + 7)
1w - 2 < -8w + 14
Now, let's combine like terms:
w - 2 < -8w + 14
To isolate the variable w, let's bring all the w terms to the left side and all the constant terms to the right side:
w + 8w < 14 + 2
9w < 16
Finally, we isolate w by dividing both sides of the inequality by 9:
(9w)/9 < 16/9
w < 16/9
Therefore, the solution to the inequality 1/2w - 1 < -4w + 7 with the variable w simplified as much as possible is w < 16/9.